CHAPTER XXVI EFFECTS OF HIGHER HARMONICS 251. To elucidate the variation in the shape of alternating waves caused by various harmonics, in Figs. 185 and 186 are shown the wave-forms produced by the superposition of the P44S4t 4i' Fig. 185. triple and the quintuple harmonic upon the fundamental sine wave. In Fig. 185 is shown the fundamental sine wave and the com- plex waves produced by the superposition of a triple harmonic of 30 per cent, the amphtude of the fundamental, under the rela- 24 369 370 AL TERN A TING-C URREN T PHENOMENA tive phase displacments of 0°, 45°, 90°, 135°, and 180°, repre- sented by the equations: sin /3 sin |8 - 0.3 sin 3 /S sin /3 - 0.3 sin (3 /3 - 45°) sin /3 - 0.3 sin (3 /3 ~ 90°) sin )8 - 0.3 sin (3 /S - 135°) sin ^ - 0.3 sin (3 ^ - 180°). ^^.^. 4i^ ft Fig. 186. As seen, the effect of the triple harmonic is, in the first figure, to flatten the zero values and point the maximum values of the wave, giving what is called a peaked wave. With increasing phase displacement of the triple harmonic, the flat zero rises and gradually changes to a second peak, giving ultimately a flat-top or even double-peaked wave with sharp zero. The intermediate positions represent what is called a saw-tooth wave. In Fig. 186 are shown the fundamental sine wave and the EFFECTS OF HIGHER HARMONICS 371 complex waves produced by superposition of a quintuple har- monic of 20 per cent, the amplitude of the fundamental, under the relative phase displacement of 0°, 45°, 90°, 135°, 180°, represented by the equations: sin /3 sin iS - 0.2 sin 5 jS sin ^ - 0.2 sin (5 /S - 45°) sin /3 - 0.2 sin (5 ^S - 90°) sin 13 - 0.2 sin (5 /S - 135°) sin 13 - 0.2 sin (5 i3 - 180°). Fig. 187. — Some characteristic wave-shapes. The quintuple harmonic causes a flat-topped or even double- peaked wave with flat zero. With increasing phase displacement the wave becomes of the type called saw-tooth wave also. The flat zero rises and becomes a third peak, while of the two former 372 ALTERNATING-CURRENT PHENOMENA peaks, one rises, the otlier decreases, and the wave gradually changes to a triple-peaked wave with one main peak, and a sharp zero. As seen, with the triple harmonic, flat top or double peak coincides with sharp zero, while the quintuple harmonic flat top or double peak coincides with flat zero. Sharp peak coincides with flat zero in the triple, with sharp zero in the quintuple harmonic. With the triple harmonic, the saw-tooth shape appearing in case of a phase difference between fundamental and harmonic is single, while with the quintuple harmonic it is double. Thus in general, from simple inspection of the wave-shape, the existence of these first harmonics can be discovered. Some characteristic shapes are shown in Fig. 187. Flat top with flat zero, sin i3 - 0.15 sin 3/3-0.10 sin 5 /3. Flat top with sharp zero, sin ^ - 0.225 sin (3 j8 - 180°) - 0.05 sin (5 /3 - 180°). Double peak, with sharp zero, sin /? - 0.15 sin (3 /? - 180°) - 0.10 sin 5 /3. Sharp peak with sharp zero, sin i8 - 0.15 sin 3 )3 - 0.10 sin (5 /3 - 180°). For further discussion of wave-shape distortion by harmonics see "Engineering Mathematics." 252. Since the distortion of the wave-shape consists in the superposition of higher harmonics, that is, waves of higher fre- quency, the phenomena taking place in a circuit supplied by such a wave will be the combined effect of the different waves. Thus in a non-inductive circuit the current and the potential difference across the different parts of the circuit are of the same shape as the impressed e.m.f. If inductive reactance is inserted in series with a non-inductive circuit, the self-inductive reactance consumes more e.m.f. of the higher harmonics, since the reactance is proportional to the frequency, and thus the current and the e.m.f. in the non-inductive part of the circuit show the higher harmonics in a reduced amplitude. That is, self-inductive react- ance in series with a non-inductive circuit reduces the higher harmonics or smooths out the wave to a closer resemblance to sine-shape. Inversely, capacity in series to a non-inductive circuit consumes less e.m.f. at higher than at lower frequency, and thus makes the higher harmonics of current and of potential EFFECTS OF HIGHER HARMONICS 373 difference in the non-inductive part of the circuit more pro- nounced— intensifies the harmonics. Self-induction and capacity in series may cause an increase of voltage due to complete or partial resonance with higher har- monics, and a discrepancy between volt-amperes and watts, without corresponding phase displacement, as will be shown hereafter. 253. In long-distance transmission over lines of noticeable inductive and condensive reactance, rise of voltage due to reso- nance may occur with higher harmonics, as waves of higher fre- quency, while the fundamental wave is usually of too low a frequency to cause resonance. An approximate estimate of the possible rise by resonance with various harmonics can be obtained by the investigation of a numerical example. Let in a long-distance line, fed by step-up transformers at 60 cycles. The resistance drop in the transformei'S at full-load = 1 per cent. The reactance voltage in the transformers at full-load = 5 per cent, with the fundamental wave. The resistance drop in the line at full-load = 10 per cent. The reactance voltage in the line at full-load = 20 per cent, with the fundamental wave. The capacity or charging current of the line = 20 per cent, of the full-load current, /, at the frequency of the fundamental. The line capacity may approximately be represented by a condenser shunted across the middle of the line. The e.m.f. at the generator terminals, E, is assumed as maintained constant. The e.m.f. consumed by the resistance of the circuit from generator terminals to condenser is Ir = OmE, or, ■pi r = 0.06 J-- The reactance e.m.f. between generator terminals and con- denser is, for the fundamental frequency, Ix = 0.15E, or, jP X = 0.15 J-; 374 ALTERNATING-CURRENT PHENOMENA thus the reactance corresponding to the frequency {2 k — 1)/ of the higher harmonic is x{2k- 1) = 0.15 (2 fc- 1) J- The capacity current at fundamental frequency is, i = 0.2 /; hence, at the frequency {2 k — 1)/, i = 0.2 (2 k - 1) e' ^' if e' = e.m.f. of the {2 k — l)th harmonic at the condenser, e = e.m.f. of the {2 k — l)th harmonic at the generator terminals. The e.m.f. at the condenser is e' = Ve' - i^r^ + *'^ (2 ^ - i; hence, substituted, e' 1 e Vl - 0.059856(2 fc- 1)^ + 0.0009(2 fc - 1)' the rise of voltage by inductive and condensive reactance. Substituting, k = 1 2 3 4 5 6 or, 2k - - 1 = 1 3 5 7 9 11 and a = 1.03 1.36 3.76 2.18 0.70 0.38 That is, the fundamental will be increased at open circuit by 3 per cent., the triple harmonic by 36 per cent., the quintuple harmonic by 276 per cent., the septuple harmonic by 118 per cent., while the still higher harmonics are reduced. The maximum possible rise will take place for that is, at a frequency / = 346, and a ^ 14.4. That is, complete resonance will appear at a frequency between quintuple and septuple harmonic, and would raise the voltage at this particular frequency 14.4-fold. If the voltage shall not exceed the impressed voltage by more than 100 per cent., even at coincidence of the maximum of the harmonic with the maximum of the fundamental. EFFECTS OF HIGHER HARMONICS 375 the triple harmonic must be less than 70 per cent, of the fundamental, the quintuple harmonic must be less than 26.5 per cent, of the fundamental, the septuple harmonic must be less than 46 per cent, of the fundamental. The voltage will not exceed twice the normal, even at a fre- quency of complete resonance with the higher harmonic, if none of the higher harmonics amounts to more than 7 per cent, of the fundamental. Herefrom it follows that the danger of resonance in high-potential lines is frequently overestimated, since the conditions assumed in this example are rather more severe than found in hnes of moderate length, the capacity current of such line very seldom reaching 20 per cent, of the main current, 254. The power developed by a complex harmonic wave in a non-inductive circuit is the sum of the powers of the individual harmonics. Thus if upon a sine wave of alternating e.m.f. higher harmonic waves are superposed, the effective e.m.f. and the power produced by this wave in a given circuit or with a given effective current are increased. In consequence hereof alterna- tors and synchronous motors of iron-clad unitooth construction — that is, machines giving waves with pronounced higher harmonics — may give with the same number of turns on the armature, and the same magnetic flux per field-pole at the same frequency, a higher output than machines built to produce sine waves. 255. This explains an apparent paradox: If in the three-phase star-connected generator with the mag- netic field constructed as shown diagrammatically in Fig. 188 the magnetic flux per pole = 4>, the number of turns in series per circuit = n, the frequency = /, the e.m.f. between any two collector rings is E = \/2Trf2n^ 10-^ . since 2 n armature turns simultaneously interlink with the magnetic flux, . The e.m.f. per armature circuit is e = \/2 7r/n«J>10-8; hence the e.m.f. between collector rings, as resultant of two e.m.fs., e, displaced by 60° from each other, is E = e\/3 = V2 7r/V3n$10-^ 376 ALTERNATING-CURRENT PHENOMENA while the same e.m.f. was found from the number of turns, the magnetic flux, and the frequency by direct calculation to be equal to 2 e; that is, the two values found for the same e.m.f. have the proportion ■\/3:2 = 1 ; 1.154. This discrepancy is due to the existence of more pronounced higher harmonics in the wave e than in the wave E = e y. s/^, which have been neglected in the formula e = ^/2Trfn^lO-\ Hence it follows that, while the e.m.f. between two collector rings in the machine shown diagrammatically in Fig. 188 is only Fig. 188. — Three-phase star-connected alternator. e X V^) ^y massing the same number of turns in one slot instead of in two slots, we get the e.m.f. 2 e, or 15.4 per cent, higher e.m.f., that is, larger output. It follows herefrom that the distorted e.m.f. wave of a unitooth alternator is produced by lesser magnetic flux per pole — that is, in general, at a lesser hysteretic loss in the armature or at higher EFFECTS OF HIGHER HARMONICS 377 efficiency — than the same effective e.m.f. would be produced with the same number of armature turns if the magnetic dispo- sition were such as to produce a sine wave. 256. Inversely, if such a distorted wave of e.m.f. is impressed upon a magnetic circuit, as, for instance, a transformer, the wave of magnetism in the primary will repeat in shape the wave of magnetism interlinked with the armature coils of the alternator, and consequently with a lesser maximum magnetic flux the same effective counter e.m.f. will be produced, that is, the same power converted in the transformer. Since the hysteretic loss in the transformer depends upon the maximum value of mag- netism, it follows that the hysteretic loss in a transformer is less with a distorted wave of a unitooth alternator than with a sine wave. 257. From another side the same problem can be approached : If upon a transformer a sine wave of e.m.f. is impressed, the wave of magnetism will be a sine wave also. If now upon the sine wave of e.m.f. higher harmonics, as sine waves of triple, quintuple, etc., frequency are superposed in such a way that the corresponding higher harmonic sine waves of magnetism do not increase the maximum value of magnetism, or even lower it by a coincidence of their negative maxima with the positive maximum of the fundamental, in this case all the power represented by these higher harmonics of e.m.f. will be transformed without an increase of the hysteretic loss, or even with a decreased hysteretic loss. Obviously, if the maximum of the higher harmonic wave of magnetism coincides with the maximum of the fundamental, and thereby makes the wave of magnetism more pointed, the hyster- etic loss will be increased more than in proportion to the in- creased power transformed, i.e., the efficiency of the transformer will be low^ered. That is, some distorted waves of e.m.f. are transformed at a lesser, some at a larger, hysteretic loss than the sine wave, if the same effective e.m.f. is impressed upon the transformer. The unitooth alternator wave and the first wave in Fig. 226 belong to the former class; the waves derived from continuous- current machines, tapped at two equidistant points of the armature, frequently, to the latter class. 258. Regarding the loss of energy by Foucault or eddy currents, this loss is not affected by distortion of wave-shape, since the 378 ALTERNATING-CURRENT PHENOMENA e.m.f. of eddy currents, like the generated e.m.f., is proportional to the secondary e.m.f.; and thus at constant impressed primary e.m.f. the power consumed by eddy currents bears a constant relation to the output of the secondary circuit, as obvious, since the division of power between the two secondary circuits— the eddy-current circuit and the useful or consumer circuit — is unaffected by wave-shape or intensity of magnetism. In high-potential lines, distorted waves whose maxima are very high above the effective values, as peaked waves, are objectionable by increasing the strain on the insulation. The striking-distance of an alternating voltage depends upon the maximum value, except at extremely high frequencies, such as oscillating discharges. In the latter, the very short duration of the voltage peak may reduce the disruptive strength, as dielectric disruption requires energy, that is, not only voltage, but time also.